If the numerator is 1, then the product of it and the denominator will be that
denominator, so choose it as the only digit greater than 1 which is a cube,
which is 8. Then the sum of numerator and denominator will still be a one-digit
number, 9. 

So one solution is .

But there another solution because 8 is also the product of 2 and 4, and 2+4=6
is a one digit number 

The only other solution is .  I agree that's not reduced to lowest terms, but 
the instructions don't state anything about it having to be reduced.

There are no more solutions because the next cube after 8 is 27, but it has
factors 3 and 9, whose sum is 12, which is not a 1-digit number.

Answers:  and 

Edwin